Hamiltonian B-series and a Lie algebra of non-rooted trees
The algebraic and combinatorial structure of the class of Runge-Kutta numerical methods for the integration of ordinary differential equations can be characterized by the study of B-series. These are series indexed by rooted trees, in such a way that both the exact and numerical flows may be expressed in terms of B-series, whereas the modified vector field associated with a numerical integrator given by a B-series can be computed just manipulating trees. The pre-Lie algebraic structure introduced in the set of rooted trees by the usual product of vector fields plays an important role not only in establishing a relationship between trees and elementary differentials, but also in other aspects such as the study of composition methods. When the vector field associated with the differential equation is Hamiltonian, it is advantageous to apply numerical schemes that can be expressed themselves as canonical (or Hamiltonian) B-series, since they automatically preserve the symplectic character of the exact flow. Hamiltonian B-series are indexed by non-rooted trees and the aim of the paper is to analyze the algebraic structure of this class of trees in the same way as with the standard rooted trees associated with schemes expressed as B-series. The main result of the paper is the introduction and characterization of a Lie bracket on non-rooted trees reflecting the Poisson bracket of the associated elementary Hamiltonians. This allows in particular to compute the modified vector field of a composition of symplectic methods within the framework of Hamiltonian B-series.
scientific article; zbMATH DE number 6677835